Spatial-Causal Geometry (SCG)

The Medium

Space is not empty. Space is a physical medium whose local state is completely described by two measurable quantities: the electric permittivity \(\varepsilon_0\) and the magnetic permeability \(\mu_0\) of free space. The local propagation speed follows directly: $$ c = \frac{1}{\sqrt{\varepsilon_0\mu_0}} $$ This is not a postulate. It is a derivation. Weber and Kohlrausch found it in a capacitor discharge in 1855 before anyone knew what it meant. Kirchhoff recognized its implication in 1857. Maxwell connected it to transverse wave propagation in 1861 and verified it against the measured speed of light. The medium announced itself through electrostatics before it had a wave equation.

\(\varepsilon_0\) is the medium's acceptance — its willingness to take a field displacement. \(\mu_0\) is the medium's recovery — its drive to restore that displacement to equilibrium. Together they set the rate at which any disturbance propagates: \(c = 1/\sqrt{\varepsilon_0\mu_0}\). Where the medium is denser — higher product — recovery is slower. Where it is thinner, faster. \(c\) is not a universal constant. It is the local recovery rate of the medium. The tables list its value as measured at Earth's surface, inside Earth's gravitational field. GPS corrects for its variation every day.

The medium's two properties combine in exactly two physically independent ways. The product \(\varepsilon_0\mu_0 = 1/c^2\) sets the local propagation speed. The ratio \(\mu_0/\varepsilon_0 = Z_0^2\) sets the local impedance. These are independent. Nature uses both freedoms. Every phenomenon in the framework is expressible in terms of one or both.

\(Z_0 \approx 376.73\ \Omega\) is the equilibrium impedance of the undisturbed medium — the state the field continuously drives toward. A local departure from \(Z_0\) is a charge. A local change in the \(\varepsilon_0\mu_0\) product is gravity. Both are the same medium, reading differently.

Every measuring instrument is made of the same medium it is measuring. A ruler's length, a clock's tick rate, a cavity's resonance — all are electromagnetic processes governed by local \(\varepsilon_0\mu_0\). When the medium changes, every instrument changes with it in exactly the proportion required to leave every local measurement of \(c\) unchanged. Variation in \(\varepsilon_0\mu_0\) is locally undetectable. It is only visible in the ratio between two environments — through a photon that has traveled from one to the other. Michelson-Morley confirmed this, correctly: a null result was the only possible outcome. It ruled out a medium with a preferred frame. The \(\varepsilon_0\mu_0\) medium has no preferred frame. MM was always consistent with the medium.

Published sources: Weber & Kohlrausch (1856); Maxwell (1865); Pound & Rebka (1959); Hallman (2026), GTD Requires Changes in ε₀μ₀; Hallman (2026), KTD Requires Velocity-Dependent ε₀μ₀.

The Closure Invariant γ_cause

Any oscillation propagating through any medium with a propagation speed constraint traces a path whose arc length exceeds the forward distance traveled. The ratio of arc length to forward distance is fixed by geometry alone — independent of frequency, amplitude, wavelength, and the nature of the medium: $$ \gamma_{\rm cause} = \frac{2}{\pi}\,E(-1) \approx 1.2160 $$ where \(E(\cdot)\) is the complete elliptic integral of the second kind. \(\gamma_{\rm cause}\) is as substrate-independent as \(\pi\). The medium inherits it. It does not generate it.

The derivation is straightforward. Two oscillations of different frequencies traveling at the same speed traverse the same distance in the same time. Causality cannot distinguish between them. Arc length per unit forward distance must therefore be the same for every frequency. For a sinusoidal oscillation \(y = A\sin(kx)\), the arc-to-closure ratio depends only on \(\beta = Ak\). For frequency-independence, \(\beta\) must be constant. The self-referential condition — no external length scale, amplitude equals the oscillation's own radian length scale — forces \(\beta = 1\), giving \(A = \lambda/2\pi = \bar{\lambda}\). This is the unique least-work closure path. Maupertuis's principle confirms it independently: the \(\beta = 1\) curve is the path that minimizes arc length per cycle subject to the speed constraint. Both derivations give the same number. One geometry.

The reduced wavelength \(\bar{\lambda} = \lambda/2\pi\) is not a quantum postulate. It is the geometric amplitude condition of any closure-constrained oscillation. Quantum mechanics has used it correctly since de Broglie without knowing why. \(\hbar = p\bar{\lambda}\) is what any bounded oscillation with a speed constraint must satisfy. It falls out of the closure condition. No mystery.

Discreteness is what a scalar field does. Only closure-satisfying geometries are stable. Structures whose geometry does not close without discontinuity disperse. What persists are the structures satisfying \(\beta = 1\). This is not quantization imposed from outside — it is the natural filter of any field that supports propagating oscillations with a speed constraint. The discreteness of atomic shells, nuclear magic numbers, and particle masses all trace to this single geometric condition.

\(\gamma_{\rm cause}\) has been confirmed at nine independent scales: photon arc geometry, particle mass via Sagnac inversion, atomic orbital radii, the Bohr radius, the fine-structure constant, vortex coherence wavelengths, galactic domain spacing in 175 SPARC galaxies (median RMSD 1.06 km/s, zero free parameters), gravitational lensing Einstein radii (186 lenses, 18% systematic improvement), and the solar coherence bubble (Pioneer and Voyager anomaly angles). The same number. One geometry. The agreements are not coincidences. They are the signature of one closure condition operating at every scale.

Published sources: Hallman (2026), γ_cause — A Geometric Closure Invariant; Hallman (2025), Photon Structure, Scale, and Interaction from First Principles.

Gravity

Gravity is a gradient, not a force.

From D1 and confirmed experiment (Pound-Rebka 1959, GPS nanosecond corrections daily): clock rates are electromagnetic process rates set by local \(\varepsilon_0\mu_0\); clock rates vary with gravitational potential; therefore \(\varepsilon_0\mu_0\) varies with gravitational potential. A structure propagating through a region where \(\varepsilon_0\mu_0\) varies experiences a bias toward higher \(\varepsilon_0\mu_0\). The only velocity scale available to a structure governed by \(\varepsilon_0\mu_0\) is \(c^2 = 1/(\varepsilon_0\mu_0)\). Therefore: $$ \mathbf{a} = c^2\,\nabla\ln(\varepsilon_0\mu_0) $$ No force. No action at a distance. A medium with a gradient and structures that follow it. In the weak-field limit this recovers Newtonian gravity exactly. Mercury's perihelion precession of 42.9 arcsec/century follows from the \(\varepsilon_0\mu_0\) field profile alone, no kinematic term, no free parameters.

Gravitational acceleration and inertial acceleration are not merely equal — they are the same phenomenon. Both are \(\nabla(\varepsilon_0\mu_0)\). Observed from outside a closed field configuration: gravity. Experienced from inside: inertia. Confirmed to one part in \(10^{15}\) by Eötvös-class experiments. At that precision it is not a principle of analogy. It is an identity. Centripetal acceleration is acceleration; a rotating frame generates its own local \(\varepsilon_0\mu_0\) depression.

Gravitational lensing is Snell's law in a graded \(\varepsilon_0\mu_0\) medium. The refractive index: $$ n = \frac{c_{\rm ref}}{c_{\rm local}} = \sqrt{\frac{(\varepsilon_0\mu_0)_{\rm local}}{(\varepsilon_0\mu_0)_{\rm ref}}} $$ The factor of 2 over the Newtonian prediction falls out of Fermat's principle in a graded medium without modification. No curved spacetime is required. The gravitational \(\varepsilon_0\mu_0\) gradient preserves \(Z_0\) — \(\varepsilon_0\) and \(\mu_0\) scale together, so their ratio is unchanged — producing refraction without reflection at any layer. Gravity is a perfectly impedance-matched graded-index optical medium.

The product rule applied to Maxwell's own Gauss's law reveals the unification of gravity and electromagnetism directly: $$ \nabla\cdot(\varepsilon_0\mathbf{E}) = 0 \;\Rightarrow\; \nabla\cdot\mathbf{E} = -\mathbf{E}\cdot\nabla\ln\varepsilon_0 $$ In any gravitational field, \(\nabla\varepsilon_0 \neq 0\) (confirmed by Pound-Rebka and GPS). Therefore every gravitational field produces a nonzero effective charge density: $$ \rho_{\rm eff} = -\varepsilon_0(\mathbf{E}\cdot\nabla\ln\varepsilon_0) $$ This is not a correction term. It is an exact algebraic identity from Maxwell, applied to a non-constant medium. Gravity is not electromagnetically neutral. It never was. The assumption of constant \(\varepsilon_0\) hid it. The gravitational potential \(V = GM/R\) is a real electromagnetic voltage — the capacitor voltage of the planetary gravitational field — confirmed by the Schumann resonance (predicted 7.49 Hz, measured 7.83 Hz), planetary lightning, and the lunar dust levitation observed by Artemis II on April 6, 2026.

Published sources: Hallman (2026), Forensic Examination of the Kinematic Term in Special and General Relativity; Hallman (2025/2026), Unified View of Charge, Neutrinos, Photons and Gravity.

Kinematic Time Dilation Is the Doppler Effect Misattributed

Every confirmed measurement attributed to kinematic time dilation (KTD) was performed inside a rotating reference frame. Rotation is centripetal acceleration. Centripetal acceleration is a local \(\varepsilon_0\mu_0\) gradient. The Sagnac effect — the correct geometric account of that gradient — produces numerically identical results in every case without invoking any clock rate property of a moving object. GPS: the full \(+45\ \mu\text{s/day}\) is gravitational; the claimed \(-7\ \mu\text{s/day}\) is the Sagnac effect of orbital rotation. Hafele-Keating: the directional asymmetry between eastward and westward clocks is the Sagnac effect of Earth's rotating frame. Muon lifetime extension: atmospheric density gradient plus storage ring centripetal acceleration. The kinematic term was never needed.

The term \(\sqrt{1-v^2/c^2}\) is a classical Doppler propagation relation — a property of the changing distance between source and receiver — absorbed into SR's invariance condition at the step where \(dx = v\,dt\) enters as \(dx^2\) and misattributed to the rate of the moving clock. The derivation contains no clock internals and identifies no physical mechanism by which velocity alone alters the rate of a clock's oscillation. The Doppler formula lives in the field between source and observer. It does not belong to the clock.

The train whistle confirms this directly. The Doppler formula for a moving whistle is structurally identical to the formula for a moving light source, with \(c_{\rm sound}\) replacing \(c\). No one attributes the whistle's pitch change to the train's clock slowing. The path geometry does the entire job, as it does for every source of waves in every medium. The SR interpretation — clock slowing as the mechanism — produces a wrong prediction for sound. The identical interpretation applied to light is wrong for the same reason. The medium is different. The formula is the same. The clock does not know which medium it is in.

KTD is algebraically inconsistent with SR's own postulates. For KTD to reduce electromagnetic process rates by \(\gamma^{-1}\), local \(\varepsilon_0\mu_0\) must increase by \(\gamma^2\). But SR's second postulate and the Lorentz transformation together establish that \(\varepsilon_0\) and \(\mu_0\) are invariant under velocity. Any two of \{Maxwell's c, KTD, SR's ε₀μ₀ invariance\} may hold. All three cannot. The contradiction is internal to the orthodox framework, algebraic, and exact.

Dark matter, the cosmological constant, and metric singularities are all downstream passengers of the kinematic term being absorbed into the Minkowski spacetime metric in 1908 and propagated into Schwarzschild in 1916. Remove the passenger and the need for each dissolves.

Published sources: Hallman (2026), Forensic Examination of the Kinematic Term in Special and General Relativity; Hallman (2026), KTD Requires Velocity-Dependent ε₀μ₀.

The Photon

A photon is a propagating geometry in the \(\varepsilon_0\mu_0\) medium. It is not a point particle and not a pure electromagnetic oscillation. It traces a type-II elliptic arc — the curve fixed by the closure condition \(\beta = Ak = 1\) from the closure geometry above — which forces the transverse amplitude to \(A = \bar\lambda = \lambda/2\pi\). The arc length of this curve over one full wavelength exceeds the wavelength itself by exactly \(\gamma_{\rm cause} \approx 1.2160\), confirmed by direct integration against the elliptic integral formula to machine precision.

The photon's total cycling energy is carried not by the curvature at any single point but by the total arc length the field traces over one full cycle: $$ E_{\rm total} = \gamma_{\rm cause}\cdot h\nu $$ This splits into two parts. The transferable interaction energy \(h\nu\) is what is exchanged at absorption — the conventional Planck quantum, exactly matched by every photoelectric and Compton measurement ever made. The persistent propagation engine \((\gamma_{\rm cause}-1)\cdot h\nu \approx 0.216\,h\nu\) is the structural overhead that keeps the oscillation alive from one apex to the next. Orthodox quantum mechanics measured only the transferable piece. It was never wrong about \(h\nu\). It was silent about the rest.

The photon's transverse radius is \(\bar\lambda\) — its geometric width fixed by the \(\beta=1\) arc condition. For hydrogen Lyman-alpha (\(\lambda \approx 121\ \text{nm}\)), the transverse radius is about 19 nm. The wave train extends up to half a meter in length. The photon is not a point.

E and B are not cause and effect. They are two simultaneous material responses of the \(\varepsilon_0\mu_0\) medium to a single disturbance — E is the permittance response (governed by \(\varepsilon_0\)), B is the reluctance response (governed by \(\mu_0\)). Both peak together, both pass through zero together. There is no lag, and therefore no route by which a propagating photon could maintain E and B genuinely out of phase. The Poynting vector \(|\mathbf{S}| = |\mathbf{E}|^2/Z_0\) pulses at twice the photon frequency. Since \(Z_0\) is invariant under gravitational gradients, energy flux is conserved along the entire path. The photon does not give energy to the medium in transit.

Frequency shifts are \(\varepsilon_0\mu_0\) ratios between emission and reception environments. The photon carries the geometry of its origin and delivers it to its destination unchanged. Cosmological redshift: $$ z + 1 = \sqrt{\frac{(\varepsilon_0\mu_0)_{\rm here}}{(\varepsilon_0\mu_0)_{\rm there}}} $$ Not expansion. Not energy loss. A field ratio. Every redshift survey ever conducted is an \(\varepsilon_0\mu_0\) gradient map of the observable universe.

Published sources: Hallman (2025), Photon Structure, Scale, and Interaction from First Principles; Hallman (2026), The Seasonal Stellar Frequency Shift Is the Sagnac Effect.

Mass and Particle Structure

Mass is what rotation costs the medium.

The Sagnac phase formula \(\Delta\phi = 4\pi A\omega/\lambda c\) is confirmed at every accessible scale — from laboratory ring interferometers to GPS satellites. A stable particle is a closed rotating field mode in the \(\varepsilon_0\mu_0\) medium. Applying the Sagnac formula at the particle scale with the closure condition \(\Delta\phi = 2\pi n\) yields: $$ \boxed{m = \frac{\gamma_{\rm cause}^2\,\hbar}{r_{\rm clos}\,c}} $$ Zero free parameters. The same equation that measures Earth's rotation in a ring interferometer determines the proton's mass. The scale changes from interferometer to nucleus. The physics does not.

From this single closure condition, five independently measured quantities emerge:

One mechanism. Five numbers. Zero free parameters. The mass ratio was not put in. It came out.

The proton is a single S¹ rotating \(\varepsilon_0\mu_0\) closure with closure radius \(r_{\rm clos}^{(p)} = 0.3110\ \text{fm}\). The electron is its conjugate closure with \(r_{\rm clos}^{(e)} = 571.1\ \text{fm}\). The electron is not a point particle — its closure radius is derivable from its mass alone, and the high-energy scattering "radius" is simply the resolution limit of the probe wavelength.

The neutron is not a fundamental particle. It is the ground-state closure of a proton-electron pair above the critical \(\varepsilon_0\mu_0\) density threshold. Below that threshold, hydrogen is the ground state; above it, the neutron is. The energy accounting is exact: \(m_p + m_e + 0.782\ \text{MeV} = m_n\). The 0.782 MeV is the depth of the energy well between the two ground states. Beta decay is the lock releasing as density falls below threshold. Electron capture is the lock forming as density rises above it. No weak force mediates this — the local \(\varepsilon_0\mu_0\) density is the complete determining condition. The antineutrino emitted in beta decay is the impedance differential of the gap field propagating outward at \(c\) as a density-wave disturbance — the same class of object as a gravitational wave, at nuclear scale.

The particle circumference satisfies \(C = 2\pi r_{\rm clos} = \gamma_{\rm cause}^2\cdot\lambda_{\rm Compton}\). \(\gamma_{\rm cause}^2\) governs closed loops; the photon's open arc carries one factor of \(\gamma_{\rm cause}\), giving a total photon Sagnac mass-energy of \(m_{\rm total} = \gamma_{\rm cause}\,h\nu/c^2\) — independently confirming the photon energy result above from a completely different derivation.

The Higgs field is \(\varepsilon_0\mu_0\). Maxwell had it in 1865. The LHC signal at 125 GeV confirmed what Maxwell wrote — that a medium described by \(\varepsilon_0\mu_0\) permeates all space and governs the propagation of every field mode. The Higgs mechanism is the \(\varepsilon_0\mu_0\) coherence threshold condition, the same geometry that governs superconductivity. Mass is what rotation costs the medium; the Higgs field is the medium itself.

Published sources: Hallman (2026), The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure; Hallman (2026), γ_cause — A Geometric Closure Invariant.

Charge and Impedance

The \(\varepsilon_0\mu_0\) medium at rest is \(Z_0\) everywhere — featureless, undeformed. Every disturbance propagates and recovers at \(c\) locally. A stable vortex closure continuously prevents recovery. The sustained departure from \(Z_0\) is charge. The particle does not have charge. The particle is charge — the departure itself, maintained against the medium's continuous drive to return to \(Z_0\). Remove the rotation and the charge disappears. The rotation is the charge.

Charge sign is the direction of the gradient: diverging above \(Z_0\) is positive; converging below is negative. \(e\) is the unit of one closure. Integer charge counts are integer closure counts. There is no quantization mystery — the unit was always the closure.

The proton's surface impedance: $$ Z_p = Z_0\,\exp\!\left(\frac{1}{2\gamma_{\rm cause}^2}\right) \approx 528.3\ \Omega $$ The electron is its exact conjugate: $$ Z_e = Z_0\,\exp\!\left(-\frac{1}{2\gamma_{\rm cause}^2}\right) \approx 268.5\ \Omega $$ These satisfy two exact conjugacy relations: \(\Gamma_p + \Gamma_e = 0\) and \(Z_p \cdot Z_e = Z_0^2\). The electron is not the proton's opposite — it is the exact impedance termination the proton's gradient requires. Charge conservation is impedance matching conservation: an open mismatch cannot exist without its conjugate. The medium cannot sustain an unbalanced gradient. This is not a law imposed on the framework. It is the self-consistency condition of the medium.

Gravity and charge are the same gradient, two projections. A product perturbation (\(\varepsilon_0\mu_0\) changes, \(Z_0\) preserved) is gravity: universal, unshieldable. A ratio perturbation (\(Z_0\) departs, propagation speed unchanged) is charge: local, shieldable. Same medium, two independent combinations, two reference scales. The gravitational potential IS the electromagnetic voltage. \(V = GM/R\) is one quantity in two unit systems.

Annihilation is not a collision — it is curl cancellation. When matter meets antimatter, the conjugate curl geometries combine to zero net departure from \(Z_0\). The field energy stored in both departures — total mass energy \(2mc^2\) — propagates outward as the medium recovers. The photons are not created in the event. They are the recovery.

Published sources: Hallman (2026), Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure; Hallman (2025/2026), Unified View of Charge, Neutrinos, Photons and Gravity.

Handedness: χ = +1

The \(\varepsilon_0\mu_0\) medium at rest has no handedness. A static scalar field has no curl, no orientation, nothing handed about it. Handedness appears precisely and only when something moves through the medium. Motion through \(\varepsilon_0\mu_0\) in three-dimensional space produces a curl with a definite, physically unambiguous orientation — the same orientation at every scale, in every experiment, for every moving thing in this universe. That orientation is right-handed. The primitive is \(\chi = +1\): not a property of the static medium, but a property of motion through it.

The handedness of a stable particle is set by which side of ambient the oscillation closes from. The \(\varepsilon_0\mu_0\) medium has a rest state — ambient. Oscillations occupy one of three topological states relative to that ambient:

• Above ambient: oscillation peak above, trough grazing ambient at the lower focus. Propagates forward. Right-handed. This is the photon.
• Below ambient: trough below, peak grazing ambient. Propagates opposite direction. Left-handed. This is the anti-photon — indistinguishable from a photon traveling the other way when read at ambient.
• At ambient: oscillation closes on itself rather than propagating. Localizes as a particle. Handedness is set by which side of ambient the oscillation came from at closure.

A closure from an above-ambient oscillation drops to ambient and winds right-handedly — this is the electron. A closure from a below-ambient oscillation rises to ambient and winds left-handedly — this is the proton. The winding direction is the memory of which side of ambient the oscillation occupied. It is baked into the topology at closure and cannot be changed by any subsequent reorientation.

The four possible pairings: above + right-handed (electron, stable), below + left-handed (proton, stable), above + left-handed (positron, unstable), below + right-handed (antiproton, unstable). The two stable pairings are those whose winding is compatible with the diverging \(\varepsilon_0\mu_0\) gradient of the ambient field — gravity. The two unstable pairings wind against the diverging grain and are geometrically eroded by it.

Matter dominance is not a fine-tuning of initial conditions or a consequence of CP violation. It is geometric selection by the ambient diverging field. Equal amounts of all four handedness states form. The two compatible with the grain of the gravitational \(\varepsilon_0\mu_0\) medium survive. The two incompatible ones do not. The universe is matter-dominated because gravity exists, and because motion through the medium is right-handed.

The electron-positron annihilation gammas confirm this directly: two 511 keV photons with opposite circular polarization. This measurement has been in the literature since the 1940s. It is the signature of opposite winding handedness releasing to opposite sides of ambient — the electron's right-handed above-ambient closure releasing as a right-handed photon, the positron's left-handed above-ambient closure releasing as a left-handed anti-photon.

Published sources: Hallman (2026), The Physical Origin of Spin and the Wrong Turn of 1925.

Atomic Structure and Spectroscopy

The Bohr radius is not a postulate. It is a geometric consequence of closure geometry. From the Sagnac closure condition and the fine-structure constant: $$ a_0 = \frac{\hbar}{m_e c\alpha} = 52{,}919\ \text{fm} $$ The fine-structure constant \(\alpha \approx 1/137.036\) is the coupling efficiency between a static charge geometry and a propagating \(\varepsilon_0\mu_0\) field cycle — a three-component geometric product with zero free parameters.

The Rydberg formula is a confinement geometry identity. The photon's reduced wavelength emitted in a transition from shell \(n_2\) to \(n_1\) is: $$ \bar{\lambda} = \frac{2\,n_1^2\,n_2^2\,a_0}{\alpha\,(n_2^2 - n_1^2)} $$ This is the inter-shell confinement geometry (the product of the two orbital radii divided by their separation), scaled by the coupling efficiency \(\alpha\), spanning the full diameter. Lyman-alpha: predicted 121,506 pm, measured 121,568 pm, error −0.051%. H-alpha: predicted 656,133 pm, measured 656,279 pm, error −0.022%. Zero free parameters throughout.

A spectral line is not a photon property. It is a record of the source geometry — the impedance traversal profile of the collapsing electron closure between two orbital states. The linewidth is the energy distribution of that collapse across the impedance gradient, derivable from the exponential profile \(Z(r) = Z_0\exp(-\tfrac{1}{2}\gamma_{\rm cause}^2\,r_{\rm clos}/r)\) evaluated between the two orbital radii. The Lorentzian line shape is a prediction of the exponential gradient, not a postulate. Every spectrometer ever built has been reading the impedance traversal profile of source closure transitions — without that identification ever being made. Orthodoxy stopped at \(\Delta E = h\nu\), matched the center frequency to an energy level table, and called it done. The data was always richer.

The presence of \(\alpha\) in the Bohr radius is not a reference to photon-electron coupling. \(\alpha\) is the shape parameter of the electron's impedance well — it sets how steeply \(Z_e(r)\) decays back toward \(Z_0\) with distance. The Bohr radius is where that gradient balances the proton's inward gravitational gradient: G pulling in, Z pushing out, equilibrium where they match. The photon couples to the electron at efficiency \(\alpha\) because the electron is already at that radius — the coupling is a consequence of the geometry, not its cause. The atom was not designed for absorption. Absorption is what the well looks like from outside. Chemistry is impedance matching. Ions are residual mismatches. Covalent bond length follows a \(\sqrt{2}\) compression: \(r_{\rm bond} = \sqrt{2}\,r_{\rm valence}\). For H\(_2\): predicted 74.84 pm, measured 74 pm, 1.1% error. H\(_2\) binding energy = \(\frac{1}{3}\) Rydberg, predicted 4.520 eV, measured 4.478 eV, 0.9% error. Zero free parameters.

Published sources: Hallman (2025), Unified View of Charge, Neutrinos, Photons and Gravity in SCG; Hallman (2026), Sagnac Formula Inverted.

Galactic Rotation

The rotation curves of galaxies require no dark matter. They follow directly from the \(\varepsilon_0\mu_0\) field geometry through a single geometric law.

The locations of kinematic transitions in galactic rotation curves — the inflection points where velocity profiles change slope — are predicted before any velocity data is consulted by: $$ \Delta r_i = \gamma_{\rm cause}\sqrt{r_i} $$ where \(r_i\) is the inner radius of domain \(i\) and \(\Delta r_i\) is its radial width. Domain boundaries are computed from the galactic center outward using only \(\gamma_{\rm cause} = 1.216\). No velocity data. No mass model. No fitted parameters.

Applied to all 175 galaxies in the SPARC database:

The median RMSD of 1.06 km/s is an order of magnitude smaller than the observational uncertainty of the rotation curves. These are not fits. They are pre-computed geometric predictions confirmed against data that had not yet been seen when the boundaries were computed.

The \(\sqrt{r}\) spacing is the geometric necessity of the closure condition at galactic scale. In a rotating disk governed by \(\mathbf{a} = c^2\nabla\ln(\varepsilon_0\mu_0)\), the natural propagation length at radius \(r\) is \(\sqrt{r}\). The constant of proportionality is \(\gamma_{\rm cause}\) — the same arc-to-closure ratio that governs photon structure. The flat rotation curve is the asymptotic behavior of the outer domain sequence. MOND's empirical acceleration threshold \(a_0\) marks the transition between the inner and outer domain curvature regimes — the geometry predicts it without introducing \(a_0\) as a free parameter. The dark matter "problem" is a misreading of domain structure in the \(\varepsilon_0\mu_0\) field.

The same invariant governs gravitational lensing. The corrected Einstein radius: $$ \theta_E^\gamma = \gamma_{\rm cause}\cdot\theta_E^{\rm GR} $$ Applied to 186 lenses from the CASTLES and SLACS catalogs: 18% systematic improvement in Einstein radius prediction, zero free parameters. The factor of \(\gamma_{\rm cause}\) is the same constant that governs photon arc geometry — confirmed at lensing scale independently.

Published sources: Hallman (2025), Galactic Rotation Curves Without Dark Matter Using SCG and the γ_cause Invariant; Hallman (2025), Gravitational Lensing Without Dark Matter Using SCG and the γ_cause Invariant.

Solar System Dynamics

The solar \(\varepsilon_0\mu_0\) field is not spherically symmetric. It is flattened by the ecliptic plane mass concentration into a bubble whose angular coherence boundary satisfies: $$ r\cdot\sin\theta = H(r) $$ where \(H(r)\) is a scale height set by \(\gamma_{\rm cause}\) and the solar field profile. A spacecraft at inclination \(\theta\) to the ecliptic crosses this coherence boundary at distance \(r_{\rm exit}\), where the smooth \(\varepsilon_0\mu_0\) gradient gives way to the interstellar background. The JPL gravitational model — which assumes spherical symmetry — records this boundary crossing as an anomalous sunward acceleration.

Pioneer 10 (\(\theta \approx 35°\)): boundary crossing at 18–22 AU. Observed anomaly onset: 18–20 AU. Predicted acceleration \(8.7\times10^{-10}\ \text{m s}^{-2}\), matching measured value. Voyager (\(\theta \approx 5°\)): boundary crossing at 110–135 AU. Observed: smooth fade beyond 100 AU, no sharp transition. The two missions showed different signatures because they crossed the same coherence boundary at different angles. No isotropic force model can produce this. Only the non-spherical bubble geometry accounts for both.

The JUICE mission's 2026 Earth gravity assist is expected to display a curvature-induced Doppler residual of approximately \(\Delta v \approx +1.7 \pm 0.3\ \text{mm s}^{-1}\), corresponding to a fractional frequency drift of \(\Delta f/f \approx 5.7\times10^{-12}\), detectable with standard DSN and ESTRACK precision. JUICE's trajectory is highly inclined (~23°) and crosses the equatorial coherence gradient near local dawn — near-optimal geometry for the test.

ʻOumuamua's non-gravitational acceleration is reproduced exactly by the same curvature-gradient mechanism, with no assumptions about composition, mass, or outgassing required.

Published sources: Hallman (2025), Solar System Dynamics under Spatial-Causal Geometry (SCG); Hallman (2026), Interstellar Object Trajectories Through the Solar ε₀μ₀ Causal-Density Bubble.

Thermodynamics and Statistical Mechanics

Heat is incoherent Sagnac mass — rotational \(\varepsilon_0\mu_0\) field depression distributed across an ensemble of closures at random phases and rates. Temperature is the average Sagnac mass per rotational degree of freedom, expressed in SI units through the unit-bridge constant \(k_B\). Entropy is \(S = k_B\ln(\varepsilon_0\mu_0)\) — not a microstate count, but the field density itself. Entropy increases because \(\varepsilon_0\mu_0\) gradients disperse, and dispersed gradients give more configurations than concentrated ones.

The arrow of time is the outward propagation of Sagnac mass transactions. Field disturbances at \(c\) do not spontaneously reconverge in an isotropic medium. The second law is not a statistical tendency. It is field geometry. Every decelerating wheel, every cooling cup of coffee, every tidal lock in progress is a Sagnac mass-change event propagating outward into the medium.

The cosmic microwave background is not a relic. It is the current thermal signature of the cosmological \(\varepsilon_0\mu_0\) medium at its equilibrium gradient state. A perfect blackbody spectrum is what a uniform field at equilibrium produces. The CMB is the medium reading its own temperature.

Superconductivity is the cooperative zero-gradient state of the conduction closures. The critical temperature \(T_c\) is the gradient threshold below which the cooperative geometry is stable against thermal disruption. Across conventional metals, type-II compounds, cuprates, hydrides, moiré systems, and marginal superconductors: zero free parameters, no material-specific mechanisms. The same \(\gamma_{\rm cause}\) closure condition that sets particle mass sets the critical temperature.

Published sources: Hallman (2025), Causal Transition Dynamics under SCG and the γ_cause Invariant.

Unification

The \(\varepsilon_0\mu_0\) framework is not a new theory of gravity, or a new theory of electromagnetism, or a new theory of quantum mechanics. It is the identification of the common substrate all three were already describing. Maxwell's equations describe the medium. Newtonian gravity is the weak-field limit of \(\mathbf{a} = c^2\nabla\ln(\varepsilon_0\mu_0)\). General relativity is the graded-index wave equation for \(\varepsilon_0\mu_0(x)\) written in metric language. Quantum mechanics is the closure geometry of the same field at small scales. These are not separate theories that need reconciling. They are the same medium seen from different angles.

One field. Two independent combinations: the product (\(\varepsilon_0\mu_0\), governing propagation speed and gravity) and the ratio (\(\mu_0/\varepsilon_0\), governing impedance and charge). One geometric invariant: \(\gamma_{\rm cause} \approx 1.2160\), substrate-independent, the arc-to-closure ratio of any propagating oscillation in any c-constrained medium. Four states of the field:

• Oscillating space — the photon.
• Spinning space — a particle.
• Dense space — gravity.
• Diverging or converging space — charge.

The physical constants are not free parameters. They are the field's own geometry expressed in human measurement units. \(c\) is the local recovery rate of the medium. \(\hbar\) is the action quantum of the closure condition in SI units. \(G\) is the unit bridge between local curvature and the background cosmic \(\varepsilon_0\mu_0\). \(k_B\) is the conversion between Sagnac mass quanta and Kelvin. All are local. Only dimensionless quantities — \(\alpha\), \(\gamma_{\rm cause}\), integer winding numbers — are genuinely universal.

Published sources: Hallman (2025), Correcting the Two False Assumptions That Fragmented Physics; Hallman (2025), Physical Constants as Derived from Spatial-Causal Geometry and the γ_cause Invariant; Hallman (2025), Hilbert's Sixth Problem Resolved through Spatial-Causal Geometry.

About

D. J. Hallman is an independent researcher and the originator of Spatial-Causal Geometry (SCG). His work developed through decades of critical study in electronics, field theory, quantum mechanics, and general relativity, driven by foundational questions rather than institutional programs. SCG emerged from a refusal to accept postulates as explanations — the Higgs mechanism without a mass prediction, kinematic time dilation without a physical mechanism, dark matter without a direct detection, the quantization of charge without a geometric cause.

The framework is built from what Maxwell, Faraday, and Weber actually measured, with no additional postulates. The geometry was always there.

All works are published open-access through Zenodo under a CC BY 4.0 license. Full list of publications at Zenodo. Contact: SCG@azfn.com

Note on terminology: the published papers listed above use ρ(x) as the name of the scalar field. ρ(x) ≡ ε₀μ₀(x) — they are the same field. The papers were written at an earlier stage of the framework's development, before the identification of ρ(x) with the electromagnetic permittivity-permeability product of the vacuum was completed. The physics is identical throughout. Readers moving between this summary and the Zenodo papers can substitute ε₀μ₀ for ρ everywhere.

Papers

All works are published open-access on Zenodo under a CC BY 4.0 license. Papers in the numbered series (0.x–8.x) form a coherent sequence; unnumbered papers are standalone results. Papers written before the identification of ρ(x) ≡ ε₀μ₀ use ρ(x) as the field name throughout — substitute ε₀μ₀ for ρ everywhere; the physics is identical.

Five Sentences

Space is a physical medium described by ε₀ and μ₀.

A particle is spinning space.

A photon is oscillating space.

Gravity is density of space.

Charge is diverging or converging space.

Every result on this page is a consequence of these five sentences.